# Questions tagged [plane-geometry]

Plane Geometry is about flat shapes like lines, circles and triangles , shapes that can be drawn on a piece of paper

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### Zero-dimensional functions in the plane

Is the following true? Conjecture. Let $\varphi:C\to [0,\infty)$ be an upper semi-continuous function, where $C\subseteq \mathbb R$ is a Cantor set. Let $X$ be a zero-dimensional subset of the graph ...
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### Generalized figures of constant width

Is it known which plane figures $Q$ can rotate touching three given circles $A$, $B$, and $C$? This question was asked by Lazar Lyusternik in 1946, there is only one reference to this paper that ...
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### Hypothesis: An injection from polygons into $SO(2) \times S_n$

I have stumbled upon a possible representation of polygons by a concise description of their behaviour under rotation. I would like to know more about it and, obviously, if it is actually a bijection. ...
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### Orientations of triples of points in the plane

Given a finite indexing-set $I$ and a collection $P = \{P_i: \ i \in I\}$ of points in the plane no three of which are collinear, let $I_{(3)}$ denote the set of ordered triples of distinct elements ...
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### Minimum area of the convex hull of the union of a parallelogram and a triangle

This question is somewhat dual to my previously stated question about Maximum area of the intersection of a parallelogram and a triangle, where the triangle and parallelogram each is assumed to be of ...
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### On 'fair bisectors' of planar convex regions

Definitions (https://www.ias.ac.in/article/fulltext/pmsc/122/03/0459-0467): Given a planar convex region $C$ (could be smooth or polygonal), an area bisector of $C$ is any line that partitions $C$ ...
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### Johnson-Lindenstrauss lemma preserves angles

Edit (April 1, 2020): I formalized the original question (from the reference below) wrong. Please see @AndreasBlass correction. Also, all the suggestions proposed so far doesn't seems to be related ...
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### Convergence in the Caratheodory sense and Hausdorff sense

Among Jordan domains, I understand that Caratheodory convergence is weaker than Hausdorff convergence. But if a sequence of Jordan domains all have rectifiable boundary whose arc length are all $L$, ...
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### How can we find n points on a plane so that as many pairs of points as possible have the same distance?

There are $n$ points on the plane, and we need to maximize the number of pairs of points which have the same Euclidean distance.
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### Find the smallest circumference of a figure containing n squares [closed]

So there's a figure which contains n squares of 1 x 1, and I have to find the smallest circumference possible. I don't know if there's an algorithm behind this, I've been stuck on this for two hours ...
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### To minimize the Hausdorff distance between convex polygonal regions

Definition: The Hausdorff distance is the greatest of all the distances from a point in one set to the closest point in the other set. Question: Given two convex polygonal regions P1 and P2 on the ...
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### Minimizing the number of segments in drawings of planar graphs

Every planar graph has at most $3n-6$ edges, where $n$ is the number of vertices. Moreover, every planar graph can be drawn with straight-line edges in the plane, without crossings. For example, for ...
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### Trapping lightrays under nonstandard reflections and/or paths

Almost every version of trapping lightrays with mirrors is either resolved---usually negatively---or open: "It is unknown whether one can construct a polygonal trap for a parallel beam of light": ...
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### How to find a point on a line that minimizes sum of distances from three given points?

Let there be given three points $x_1, x_2, x_3$ and a line $l$ on a plane. Does there exist an explicit method of finding such a point $p$ on $l$, that sum of distances of $p$ from $x_1, x_2, x_3$ is ...
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### What is the symmetry group of this configuration?

This configuration appear as problem 3845 in Crux Mathematicorum. I see it is very beautiful. This configuration are generalization of Pascal theorem and Brianchon theorem: Consider six points $A_1$, ...
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### On non-convex polygons that tile convex polygons

Polyominoes are rectilinear polygons - all angles are 90 or 270 degrees. It is well-known that among non-convex polyominoes are several whose copies can neatly tile rectangle. And for several such '...
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### Two queries on triangles, the sides of which have rational lengths

Let us define a "rational triangle" as one in the Euclidean plane, with lengths of all sides rational. We are aware that a positive integer is called "congruent" only if it is the area of a right ...
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### Visibility in a growing orchard

This is a variant on Polya's orchard problem.1,2 Suppose trees are planted randomly in the plane. The question is: How many trees are visible from the origin as their radii grow? More precisely, ...
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### Graphs determined by monohedral, edge-to-edge tilings of the plane

Let $\cal T$ be a monohedral, edge-to-edge tiling of the plane, with prototile $T$ a simple polygon, and with one edge $e^*$ of $T$ distinguished. Associate a graph $G=G_{\cal T}$ with $\cal T$ as ...
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### Geometric dissection theory

A few days ago, i realized that one way to prove the Pythagorean Theorem is to dissect the given right-angled triangle into 2 triangles similar to it, and apply well-known properties of ratios of ...
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### Hyperbolic Intercept (Thales) Theorem

Is there an Intercept theorem (from Thales, but don't mistake it with the Thales theorem in a circle) in hyperbolic geometry? Euclidean Intercept Theorem: Let S,A,B,C,D be 5 points, such that SA, SC, ...